The magnitude of an earthquake is rapidly estimated from the frequency content of the first four seconds of the P-wave arrival. The predominant period, , of the vertical component waveform is calculated using the method first described by Nakamura [1988]. The maximum observed within 4 sec,
, has been measured for earthquakes 3.0 M 8.3 from around the world and is found to scale with earthquake magnitude (Figure 13.22). The data shows no evidence to suggest that the scaling relation breaks down for the largest magnitude events with rupture durations greater than 4 sec.

Figure 13.22:
Scaling relation between event-averaged
and magnitude. All data has been processed using the same recursive algorithms. A) Southern California earthquakes and best fit relation (solid line). From Allen and Kanamori (2003). B) Earthquakes in Japan and best fit relation (solid line). The dashed line is the best fit relation for California shown in A which is nearly identical. From Lockman and Allen (in review). C) Global compilation of earthquakes including southern California, Japan, Taiwan and the Denali earthquake. Waveforms are a mixture of accelerometers and broadband velocity instruments.

The accuracy and timeliness of magnitude estimates are central to the usefulness of an early warning system. The accuracy of magnitude estimates are a function of the number of stations providing P-wave data. Datasets from both southern California and Japan show that using just the closest station to the epicenter the average magnitude error is 0.75 magnitude units, once data from the closest 2 stations is available the error drops to 0.6, and to 0.5 magnitude units once 4 stations provide data. The offline tests of 32 earthquakes from southern California show that magnitude estimates are available for 56% of earthquakes at the time of the S-arrival at the epicenter with an average magnitude error of 0.44 magnitude units (Figure 13.20a).